Various physics applications motivate study of Finsler geometry in the pseudo-Euclidean domain. The well-known example from classical physics is the Randers metric that can be viewed as an attempt of geometrization of the electric charges motion.
Possible pseudo-Finsler modifications of gravity were discussed in [1]}. Also in quantum field theories, Finsler's geometry is a rather popular construction. In particular, in [2]}, [3]} pseudo-Finsler geometry is applied as a Lorentz symmetry violating model. In [4]} Finsler-type modification of the Coulomb law and corresponding additional splitting of hydrogen energy levels was calculated. This result provides a way for an experimental verification of Finsler's structure.
Another example is the covariant premetric dispersion relation in electromagnetism [5]} that is presented as a general fourth order polynomial equation. Due to the well-observed birefringence effect in anisotropic optics, the covariant premetric electromagnetism must be considered as a viable theory. Remarkably, the covariant premetric dispersion relation has exactly the same form as the one exemplified by Riemann in Eq.(REF ).
